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I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it be finite? Is there any structure theorem for this kind of groups?

For the larger class of torsion compact groups structure theorems exist. In particular, They tell us that these groups are profinite. So, locally finite compact groups are profinite as well. But, can one prove more?

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  • $\begingroup$ The topological analogue of LF for locally compact group is called (locally) elliptic; it means every (finite/compact) subset is contained in a compact open subgroup. If the group is abstractly locally finite, this is even stronger, but shows the study reduces to understanding which profinite groups are locally finite as abstract group. Right now I don't remember if there are other examples than the obvious ones (namely closed subgroups of products of finite groups of bounded size). $\endgroup$
    – YCor
    Commented Feb 26, 2014 at 17:20
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    $\begingroup$ oh btw I misread since you assume compact from the beginning. Anyway the question remains: are there any examples other than closed subgroups of products of finite groups of bounded order. $\endgroup$
    – YCor
    Commented Feb 26, 2014 at 17:21
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    $\begingroup$ This question is related: mathoverflow.net/a/459/1345 $\endgroup$
    – Ian Agol
    Commented Feb 26, 2014 at 18:14
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    $\begingroup$ @Benjamin In Agol's link the assumption is weaker, namely that every element has finite order. Unless you can easily prove that a torsion profinite group is locally finite, the current question is a priori easier. $\endgroup$
    – YCor
    Commented Feb 26, 2014 at 23:08
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    $\begingroup$ I checked twice: it's a deep theorem of Zelmanov (improving the solution of the restricted Burnside theorem) that every periodic profinite group is locally finite. (see E. Zelmanov, On periodic compact groups, Israel J. Math, 77 1992 83-95.) Thus the question indeed boils down to the post linked by Agol (although I wouldn't call it be called a duplicate). $\endgroup$
    – YCor
    Commented Feb 26, 2014 at 23:49

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To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite torsion groups are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).

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  • $\begingroup$ @Agol: you've missed part of the comments, namely Zelmanov's result that periodic (not necessarily bounded exponent) profinite groups are locally finite. $\endgroup$
    – YCor
    Commented Feb 27, 2014 at 10:12
  • $\begingroup$ Ok, I fixed that. In any case, it appears that the question reduces to whether there are infinite torsion pro-p groups. $\endgroup$
    – Ian Agol
    Commented Feb 27, 2014 at 18:47
  • $\begingroup$ @Agol: OK (I was confused by "which suffices in the f.g. case since once the problem is reduced to a locally finite group, the f.g. case is trivial, but I checked the RZ book and Wilson makes no f.g. assumption) $\endgroup$
    – YCor
    Commented Feb 27, 2014 at 19:25

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